Average Error: 16.5 → 3.7
Time: 14.7s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sqrt[3]{{\left(\sin \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right)}^{3}}\right)\right) \cdot R\]
\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R
↓
\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sqrt[3]{{\left(\sin \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right)}^{3}}\right)\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r27189 = phi1;
        double r27190 = sin(r27189);
        double r27191 = phi2;
        double r27192 = sin(r27191);
        double r27193 = r27190 * r27192;
        double r27194 = cos(r27189);
        double r27195 = cos(r27191);
        double r27196 = r27194 * r27195;
        double r27197 = lambda1;
        double r27198 = lambda2;
        double r27199 = r27197 - r27198;
        double r27200 = cos(r27199);
        double r27201 = r27196 * r27200;
        double r27202 = r27193 + r27201;
        double r27203 = acos(r27202);
        double r27204 = R;
        double r27205 = r27203 * r27204;
        return r27205;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r27206 = phi1;
        double r27207 = sin(r27206);
        double r27208 = phi2;
        double r27209 = sin(r27208);
        double r27210 = r27207 * r27209;
        double r27211 = cos(r27206);
        double r27212 = cos(r27208);
        double r27213 = r27211 * r27212;
        double r27214 = lambda1;
        double r27215 = cos(r27214);
        double r27216 = lambda2;
        double r27217 = cos(r27216);
        double r27218 = r27215 * r27217;
        double r27219 = sin(r27214);
        double r27220 = -1.0;
        double r27221 = r27220 * r27216;
        double r27222 = sin(r27221);
        double r27223 = r27219 * r27222;
        double r27224 = 3.0;
        double r27225 = pow(r27223, r27224);
        double r27226 = cbrt(r27225);
        double r27227 = r27218 - r27226;
        double r27228 = r27213 * r27227;
        double r27229 = r27210 + r27228;
        double r27230 = acos(r27229);
        double r27231 = R;
        double r27232 = r27230 * r27231;
        return r27232;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 16.5

    \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
  2. Using strategy rm

  3. Applied sub-neg16.5

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right) \cdot R\]

  4. Applied cos-sum3.7

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \left(-\lambda_2\right) - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}\right) \cdot R\]

  5. Simplified3.7

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_1 \cdot \cos \lambda_2} - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right) \cdot R\]
  6. Using strategy rm

  7. Applied add-cbrt-cube3.7

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \color{blue}{\sqrt[3]{\left(\sin \left(-\lambda_2\right) \cdot \sin \left(-\lambda_2\right)\right) \cdot \sin \left(-\lambda_2\right)}}\right)\right) \cdot R\]

  8. Applied add-cbrt-cube3.7

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \color{blue}{\sqrt[3]{\left(\sin \lambda_1 \cdot \sin \lambda_1\right) \cdot \sin \lambda_1}} \cdot \sqrt[3]{\left(\sin \left(-\lambda_2\right) \cdot \sin \left(-\lambda_2\right)\right) \cdot \sin \left(-\lambda_2\right)}\right)\right) \cdot R\]

  9. Applied cbrt-unprod3.7

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \color{blue}{\sqrt[3]{\left(\left(\sin \lambda_1 \cdot \sin \lambda_1\right) \cdot \sin \lambda_1\right) \cdot \left(\left(\sin \left(-\lambda_2\right) \cdot \sin \left(-\lambda_2\right)\right) \cdot \sin \left(-\lambda_2\right)\right)}}\right)\right) \cdot R\]

  10. Simplified3.7

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sqrt[3]{\color{blue}{{\left(\sin \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right)}^{3}}}\right)\right) \cdot R\]

  11. Final simplification3.7

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sqrt[3]{{\left(\sin \lambda_1 \cdot \sin \left(-1 \cdot \lambda_2\right)\right)}^{3}}\right)\right) \cdot R\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  :precision binary64
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))